Question: $\overline{AB}$ = $2\sqrt{34}$ $\overline{BC} = {?}$ $A$ $C$ $B$ $2\sqrt{34}$ $?$ $ \sin( \angle BAC ) = \frac{3\sqrt{34} }{34}, \cos( \angle BAC ) = \frac{5\sqrt{34} }{34}, \tan( \angle BAC ) = \dfrac{3}{5}$
Solution: $\overline{AB}$ is the hypotenuse $\overline{BC}$ is opposite to $\angle BAC$ SOH CAH TOA We know the hypotenuse and need to solve for the opposite side so we can use the sine function (SOH) $ \sin( \angle BAC ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\overline{BC}}{\overline{AB}}= \frac{\overline{BC}}{2\sqrt{34}} $ $ \overline{BC}=2\sqrt{34} \cdot \sin( \angle BAC ) = 2\sqrt{34} \cdot \frac{3\sqrt{34} }{34} = 6$